which is equivalent to the requirement that the corresponding nerve functor is fully faithful (in other words, if ii is inclusion then SS is a left adequate subcategory of CC in terminology of [Isbell 1960]). The nerve functor may be viewed as a singular functor? of the functor ii.

Definition

of Cat on non-empty finite linear orders regarded as categoris, meaning that the object [n]∈Obj(Δ)[n] \in Obj(\Delta) may be identified with the category [n]={0→1→2→⋯→n}[n] = \{0 \to 1 \to 2 \to \cdots \to n\}. The morphisms of Δ\Delta are all functors between these total linear categories.

where Cat is regarded as a 1-category with objects locally small categories, and morphisms being functors between these.

So the set N(𝒞)nN(\mathcal{C})_n of nn-simplices of the nerve is the set of functors {0→1→⋯→n}→𝒞\{0 \to 1 \to \cdots \to n\} \to \mathcal{C}. This is clearly the same as the set of sequences of composable morphisms in DD of length nn:

Examples

Example

(bar construction)

Let AA be a monoid (for instance a group) and write BA\mathbf{B} A for the corresponding one-object category with Mor(BA)=AMor(\mathbf{B} A) = A. Then the nerve N(BA)N(\mathbf{B} A) of BA\mathbf{B}A is the simplicial set which is the usual bar construction of AA

N(BA)=(⋯A×A×A→→→A×A→→A→*)
N(\mathbf{B}A)
=
\left(
\cdots
A \times A \times A \stackrel{\to}{\stackrel{\to}{\to}}
A \times A \stackrel{\to}{\to} A \to {*}
\right)

It suggests the sense that a Kan complex models an ∞-groupoid. The possible lack of uniqueness of fillers in general gives the ‘weakness’ needed, whilst the lack of a coskeletal property requirement means that the homotopy type it represents has enough generality, not being constrained to be a 1-type.

Unwrapping this definition inductively in (n+m)(n+m), this says that a simplicial set is the nerve of a category if and only if all its cells in degree ≥2\geq 2 are unique compositors, associators, pentagonators, etc of composition of 1-morphisms. No non-trivial such structure cells appear and no further higher cells appear.

Historical note

The notion of the nerve of a category seems to be due to Grothendieck, which is in turn based on the nerve of a covering from 1926 work of P. S. Alexandroff?. One of the first papers to consider the properties of the nerve and to apply it to problems in algebraic topology was